X-ray reflectometry (XRR) is a well-known technique for measuring the thickness, density and surface quality of thin film layers deposited on a substrate. X-ray reflectometers typically operate by irradiating a sample with a beam of X-rays at grazing incidence, i.e., at a small angle relative to the surface of the sample, near the total external reflection angle of the sample material. Measurement of X-ray intensity reflected from the sample as a function of angle gives a profile of interference fringes, which is analyzed to determine the properties of the film layers responsible for creating the fringe profile. Several X-ray reflectometers have been described in the patent literature, such as U.S. Pat. Nos. 6,512,814, 5,619,548 and 5,923,720, whose disclosures are incorporated herein by reference.
Various methods have been developed for analyzing measured interference profiles and fitting them to simulated models, as will be explained in detail hereinbelow. Some model fitting methods use Fourier transform analysis, particularly for measuring sample thickness. For example, U.S. Pat. No. 6,754,305, whose disclosure is incorporated herein by reference, describes a method for finding the layer thicknesses of a wafer using a Fourier transform analysis. U.S. Pat. No. 5,740,226, whose disclosure is incorporated herein by reference, describes a film thickness measuring method comprising the steps of measuring reflectance of X-rays on a film, extracting interference oscillations from the measured X-ray reflectance, and Fourier transforming the interference oscillations to compute a film thickness of the film.
Some of the proposed methods for fitting the model to the measured data involve Genetic Algorithms (GA) or Evolutionary Algorithms (EA). A genetic algorithm is an optimization algorithm based on the mechanisms of evolution which uses random mutation, crossover and natural selection procedures to “breed” better models or solutions from an initial condition. For example, U.S. Pat. No. 6,192,103, whose disclosure is incorporated herein by reference, describes the use of evolutionary algorithms to find a global solution to the fitting of experimental X-ray scattering data to simulated models.
Dane et al. describe the use of known genetic algorithms for the characterization of materials in a paper entitled “Application of Genetic Algorithms for Characterization of Thin Layered Materials by Glancing Incidence X-Ray Reflectometry,” Physica B, volume 253 (1998), pages 254–268, which is incorporated herein by reference. The genetic algorithm is used during the process of comparing two x-ray profiles and modifying of a calculated profile. The authors state that the proposed genetic algorithm is able to find good fits within a single run, reducing the amount of human effort and expertise required for analysis.
Other model fitting methods known in the art employ exhaustive searching. For example, U.S. Pat. No. 6,192,103, cited above, describes an approach known in the art, in which the parameter space is divided into small, but finite, regions. An error function is calculated for each region, and the region that produces the smallest error value is chosen as the best-fit parameter vector. In a related approach, mentioned in the same patent, known as the Monte Carlo method, the parameter space is again divided into small regions. The regions are selected at random, and the error function is evaluated for each. After a certain number of regions have been chosen, or when the error value is smaller than a specified value, the search is stopped. The region with the smallest error value is chosen as the best fit.
U.S. Pat. No. 6,823,043, whose disclosure is incorporated herein by reference, describes a method for determining parameters of a material by fitting a model to an experimental X-ray scattering profile. Fitting is performed on a selected sub-range of the scattering profile and gradually extended to cover the entire profile. Several fitting methods are proposed, including a genetic algorithm.
Other model fitting methods known in the art are based on gradient methods, such as the Levenberg-Marquardt method. For example, U.S. Pat. Nos. 6,754,305 and 6,512,814, cited above, describe the use of the Levenberg-Marquardt method for XRR model fitting.